The levels of uncertainty, types of intervals

What are we uncertain about?

• Parameters prior to data (prior)
• Parameters after data (posterior)

Relating parameters to expectations or potential observtions is hard

• complex models
  • What does \(k\) mean or do in the equation: \(ks \ln\left(1+\frac{\beta I}{k}\right)\)?
• interactions between parameters (“levels” of predictors)
  • What does an interaction between dose and strain mean in a logistic regression? How does this change if infection is likely or unlikely?
• unintuitive distributions
  • What does the shape parameter, \(\alpha\) do in a Gamma distribution?

Parameter uncertainty prior to data

This (un)certainty is reflected in prior distributions

What priors say about the distribution of (unobserved) data can be hard to grok

Parameter uncertainty prior to data

What priors say about the distribution of (unobserved) data can be hard to grok

\(\therefore\) use prior predictive simulation

Parameter uncertainty after data: posterior distributions

To the extent parameters are meaningful/interpretable (e.g., a slope) you can:

1. Plot/consider the whole (marginal) distribution
2. Use intervals to highlight parameter values most consistent with data (and model and prior)

Parameter uncertainty after data: posterior distributions

2. Use intervals to highlight parameter values most consistent with data (and model and prior)
   • Intervals of defined boundaries (e.g., \(p>0.5\))
   • Intervals of defined mass / “compatibility intervals”
     • Percentile intervals (PI) have the same mass in each tail
     • Highest posterior density intervals (HPDI) are the narrowest interval with a given mass

Uncertainty in expecations after data

What parameters say about expectations on the response scale can be hard to grok

Uncertainty in expecations after data

What parameters say about expectations on the responses scale can be hard to grok

\(\therefore\) plot posterior inference (e.g., against data)

Uncertainty about future data

Involves parameter uncertainty + sampling noise.

What they say about the distribution of (future) data can be hard to grok

Uncertainty about future data

What they say about the distribution of (future) data can be hard to grok

\(\therefore\) use posterior predictive simulation

Which graph or interval to use?

As always, the key questions is what do you want to know?

Just be clear about the uncertainty you are or are not showing/including